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Using coupied variational principle, predicting fatigue crack initiation life of plane stress state plate-like component which leads to large range damage is analysed. The closed form solution about predicting fatigue crack initiation life of plane stress state plate-like component which leads to large range damage is derived. The method adopted in this paper is definite in mechanical concept，it can be widely used in analysis of predicting fatigue crack initiation life of various plate-like component which leads to large range damage. A new thinking and new method is provided for predicting crack initiation life of plate-like component.

By using of closed-form solution for predicting fatigue crack initiation life of a beam subjected to the transverse bending load in large range damage， fatigue crack initiation life of backup roll of four high mill is predicted. The method adopted in this paper is simple and effective. A new method is provided for predicting fatigue crack initiation life backup roll of four high mill.